Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Pizza values A vender sells two sizes of pizza by the slice. The small slice is \(\frac{1}{6}\) of a circular 18 -inch-diameter pizza, and it sells for \(\$ 2.00\). The large slice is \(\frac{1}{8}\) of a circular 26 -inchdiameter pizza, and it sells for \(\$ 3.00\). Which slice provides more pizza per dollar?

Step-by-Step Solution

Answer

The large slice provides more pizza per dollar.

Step 1: Calculate Area of Small Pizza

The formula for the area of a circle is \( A = \pi r^2 \). First, find the radius of the small pizza, which is half of the diameter: \( r = \frac{18}{2} = 9 \). So the area of the small pizza is \( A = \pi \times 9^2 = 81\pi \).

Step 2: Find Area of a Small Slice

The small slice is \( \frac{1}{6} \) of the entire small pizza. Therefore, the area of the small slice is \( \frac{81\pi}{6} = 13.5\pi \).

Step 3: Calculate Dollars Per Square Inch for Small Slice

The cost of the small slice is \$2.00. Calculate the pizza area per dollar by \( \frac{13.5\pi}{2} \approx 6.75\pi \) square inches per dollar.

Step 4: Calculate Area of Large Pizza

The radius of the large pizza is \( r = \frac{26}{2} = 13 \). So the area of the large pizza is \( A = \pi \times 13^2 = 169\pi \).

Step 5: Find Area of a Large Slice

The large slice is \( \frac{1}{8} \) of the entire large pizza. Therefore, the area of the large slice is \( \frac{169\pi}{8} \approx 21.125\pi \).

Step 6: Calculate Dollars Per Square Inch for Large Slice

The cost of the large slice is \$3.00. Calculate the pizza area per dollar by \( \frac{21.125\pi}{3} \approx 7.04\pi \) square inches per dollar.

Step 7: Compare the Values

Compare \( 6.75\pi \) and \( 7.04\pi \) square inches per dollar. Since \( 7.04\pi \) is greater, the large slice offers more pizza per dollar.

Key Concepts

Circle Area

The area of a circle is a fundamental concept in geometry that helps us determine the space inside the circle's boundaries. To calculate the area, we use the formula \( A = \pi r^2 \). Here, \( \pi \) is approximately 3.14159, and \( r \) represents the radius of the circle, which is half the length of the circle's diameter. For instance, if you have a pizza with an 18-inch diameter, the radius is \( 18/2 = 9 \) inches.

By plugging this radius into the formula, you calculate the area: \( A = \pi \times 9^2 = 81\pi \). This tells us the total area of the pizza when it's unsliced. Similar steps follow for any circle, so understanding how to calculate the area of a circle is crucial for tasks like comparing pizza slices.

Radius Calculation

The radius of a circle is vital in many mathematical exercises as it serves as a stepping stone to calculating the circle's area. It is the distance from the center of the circle to any point on its edge. When given the diameter of a circle, finding the radius is straightforward. Simply divide the diameter by two.

Let's say we have a pizza with a 26-inch diameter. The radius would be \( 26/2 \), which equals 13 inches. This radius is then used in the area formula \( A = \pi r^2 \) to calculate how much space the pizza covers. Always confirm the diameter before calculating, and remember this basic but essential step in geometry.

Value Comparison

Value comparison in mathematics often involves understanding how much value is received per unit of cost or other criteria. In this exercise set with pizzas, we're finding out which slice offers more pizza area per dollar. It's an important skill because it affects purchasing decisions and resource management.

By calculating the area provided per dollar spent, we determine the efficiency of the purchase. For example, the small slice of pizza has a certain dollar-per-square-inch area. By dividing the slice's area by its dollar cost, you find how efficient each dollar is spent. Contrasting these results with those from other slices or items helps identify the best deal.

• Small pizza slice area per dollar: \( \frac{13.5\pi}{2} \)
• Large pizza slice area per dollar: \( \frac{21.125\pi}{3} \)

Comparing these values, the larger slice provides more area per dollar, making it the better option if you seek more pizza for your money.

Mathematical Problem Solving

Mathematical problem solving is about breaking down a problem into manageable steps and using logical reasoning and calculation skills. It requires understanding the problem, determining what you know and what you need to find, and choosing the right tools to find the solution.

In this pizza exercise, the key steps involve:

• Calculating the entire pizza area.
• Determining the area of one pizza slice.
• Figuring out the cost per area unit for each slice.
• Comparing these values to choose the best option.

The goal is optimization, finding which option provides the better value or result. This skill aids in making informed decisions across various aspects of everyday life and lays a strong foundation for tackling more complex mathematical tasks.